Abstract
It is well known that for every bounded operator A in $L(H)$ , there exists a compact operator K in $K(H)$ such that the Weyl spectrum $\sigma_{W}(A)$ of the operator A coincides with the spectrum $\sigma(A+K)$ of the perturbed operator A. In this work, we show the extension of this relation by the use of Kato’s decomposition to the set of semi-Fredholm operators.
Highlights
In Schechter demonstrated in [ ] this result that for every bounded operator A in L(H), there exists a compact operator K in K(H) such that the Weyl spectrum σW (A) of the operator A coincides with the spectrum σ (A + K) of the perturbed operator A for a compact perturbation of Fredholm operators of index zero
This work is intended to extend this result to semi-Fredholm operators of any index using the decomposition of Kato, an extension already done by Apostol in [ ] and Herrero in [ ]
As R(B) is closed, we deduce that B is left invertible
Summary
In Schechter demonstrated in [ ] this result that for every bounded operator A in L(H), there exists a compact operator K in K(H) such that the Weyl spectrum σW (A) of the operator A coincides with the spectrum σ (A + K) of the perturbed operator A for a compact perturbation of Fredholm operators of index zero. Definition The operator A ∈ C(H) is called a semi-Fredholm operator, denoted A ∈ SF(H), if the following conditions are satisfied: ( ) R(A) is a closed subspace in H, ( ) min ind(A) = {dim N(A), dim N(A∗)} < ∞. Definition Let A ∈ C(H); the complex set ρe(A) of the operator A given by ρe(A) = μ ∈ C, (A – μI) ∈ SF(H) , will be called the essential resolvent
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
